Constructing algebras from a vector space

When applied to two copies of the same vector space \({V}\), the tensor product is sometimes called the “outer product,” since it is a linear map from two vectors to a “bigger” object “outside” \({V}\), as opposed to the inner product, which is a linear map from two vectors to a “smaller” object “inside” \({V}\).

The term “outer product” also sometimes refers to the exterior product \({V\wedge V}\) (AKA wedge product, Grassmann product), which is defined to be the tensor product \({V\otimes V}\) modulo the relation \({v\wedge v\equiv0}\). By considering the quantity \({(v_{1}+v_{2})\wedge(v_{1}+v_{2})}\), this immediately leads to the equivalent requirement of identifying anti-symmetric elements, \({v_{1}\wedge v_{2}\equiv-v_{2}\wedge v_{1}}\).

These outer products can be applied to a vector space \({V}\) to generate a larger vector space, which is then an algebra under the outer product. If \({V}\) has a pseudo inner product defined on it, we can then generalize it to the larger algebra. In the following, we will limit our discussion to finite-dimensional real vector spaces \({V=\mathbb{R}^{n}}\); generalization to complex scalars is straightforward.

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